00-321 Oleg Safronov

The discrete spectrum of the Schr\"odinger operator in the large coupling constant limit (238K, Postscript) Aug 24, 00

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Abstract. In a simple quantum mechanical model for impurities in a crystal, one studies the spectrum of a periodic Schr\"odinger operator $A=-\Delta+p(x)$, perturbed by a decaying potential $-\alpha V_0$, which models the impurity ($\alpha>0$ is a coupling constant). The basic assumptions are that $H$ has a spectral gap $\Lambda$ in its essential spectrum; eigenvalues of $A(\alpha)=A-\alpha V_0$ inside $\Lambda$ correspond to the eigenstates of the electrons observed in solid state physics. We study the number of eigenvalues of $A(\alpha)$ inside an interval $(\lambda_1,\lambda_2)\subset \Lambda$. The main purpose of our study is to obtain an asymptotics of this number, as $\alpha\to \infty$. We point out that the problem has been solved only for one-dimensional Schr\"odinger operators in \cite{Sob}. The results of the present paper deal with multidimensional Schr\"odinger operators. However we consider only the case of spherically symmetric potentials $V_0$.

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